A refinement of Wilf-equivalence for patterns of length 4

In their paper \cite{DokosDwyer:Permutat12}, Dokos et al. conjecture that the major index statistic is equidistributed among 1423-avoiding, 2413-avoiding, and 2314-avoiding permutations. In this paper we confirm this conjecture by constructing two major index preserving bijections, $\Theta:S_n(1423)\to S_n(2413)$ and $\Omega:S_n(2314)\to S_n(2413)$. In fact, we show that $\Theta$ (respectively, $\Omega$) preserves numerous other statistics including the descent set, right-to-left maximum (respectively, left-to-right minimum), and a new statistic we call top-steps (respectively, bottom-steps).